The James Clerk Maxwell Telescope Nearby Galaxies Legacy Survey (NGLS) comprises an H i-selected sample of 155 galaxies spanning all morphological types with distances less than 25 Mpc. We describe the scientific goals of the survey, the sample selection and the observing strategy. We also present an atlas and analysis of the CO J=3 - 2 maps for the 47 galaxies in the NGLS which are also part of the Spitzer Infrared Nearby Galaxies Survey. We find a wide range of molecular gas mass fractions in the galaxies in this sample and explore the correlation of the far-infrared luminosity, which traces star formation, with the CO luminosity, which traces the molecular gas mass. By comparing the NGLS data with merging galaxies at low and high redshift, which have also been observed in the CO J=3 - 2 line, we show that the correlation of far-infrared and CO luminosity shows a significant trend with luminosity. This trend is consistent with a molecular gas depletion time which is more than an order of magnitude faster in the merger galaxies than in nearby normal galaxies. We also find a strong correlation of the L[subscript: FIR]/L[subscript: CO(3-2)] ratio with the atomic-to-molecular gas mass ratio. This correlation suggests that some of the far-infrared emission originates from dust associated with atomic gas and that its contribution is particularly important in galaxies where most of the gas is in the atomic phase.